Further Results on Transverse Feedback Linearization of Multi-Input ...

Further Results on Transverse Feedback Linearization of Multi-Input Systems Christopher Nielsen and Manfredi Maggiore Abstract— In this paper we continue research on the transverse feedback linearization problem. In [1] we found sufficient conditions for this problem to be solvable. Here we present necessary and sufficient conditions for local transverse feedback linearization.

II. N OTATION AND P ROBLEM S TATEMENT Consider a control system modeled by equations of the form m X gi (x)ui =: f (x) + g(x)u. (1) x˙ = f (x) + i=1

I. I NTRODUCTION The transverse feedback linearization problem (TFLP) was formulated in [2] by Banaszuk and Hauser. Given a controlled invariant manifold Γ⋆ embedded in the state space, this problem entails finding a coordinate and feedback transformation putting the dynamics transverse to Γ⋆ into linear controllable form. When feasible, transverse feedback linearization can simplify set stabilization problems. Often, control objectives will dictate that the controller stabilize sets, rather than equilibria. In [3], for instance, the solution of a set stabilization problem is central to controlling bi-pedal locomotion. The “virtual constraints” technique in [4], used to stabilize oscillatory modes in Euler-Lagrange systems, relies on feedback linearization to stabilize an invariant set. The work of Banaszuk and Hauser in [2] characterized the solution to TFLP for single-input systems when Γ⋆ is a diffeomorphic to the unit circle. In [5] we generalized Banaszuk and Hauser’s results to the case when Γ⋆ is ⋆ diffeomorphic to the generalized cylinder Rn −k ×Tk , where k T is the k-torus. In [1] we gave sufficient conditions to solve TFLP for multi-input systems. Theorem III.1 in [1], concerning the global solution to TFLP, contains a mistake. The theorem claims to give sufficient conditions for TFLP to be globally solvable under the assumption that Γ⋆ is a contractible set. Contractibility may not, in fact, be enough to guarantee the existence of a global solution to TFLP and hence Theorem III.1 in [1] must be considered a local result. In this paper we further generalize the results of [1]. Our main result is Theorem V.1, which gives necessary and sufficient conditions for the existence of a local solution to TFLP. The global problem needs further investigation. This research was supported by the National Sciences and Engineering Research Council of Canada. Chris Nielsen was partially supported by the Ontario Graduate Scholarship in Science and Technology. The authors are with the Department of Electrical and Computer Engineering, University of Toronto, 10 King’s College Road, Toronto, ON, M5S 3G4, Canada. {nielsen,

maggiore}@control.toronto.edu

n

Here x ∈ R is the state, and u = (u1 , . . . , um ) ∈ Rm is the control input. The vector fields f, g1 , . . . , gm : Rn → T Rn are smooth (C ∞ ). We assume throughout this paper that g1 , . . . , gm are linearly independent. A. Notation If k is a positive integer, k denotes the set of integers {0, 1, . . . , k − 1}. We let col(x1 , . . . , xk ) := [x1 . . . xn ]⊤ and, given two column vectors a and b, we let col(a, b) := [a⊤ b⊤ ]⊤ . Throughout this paper by a manifold is meant a smooth manifold and by a submanifold is meant an embedded submanifold. All objects are presumed to be smooth. On a manifold M , V (M ) will denote the set of all smooth vector fields on M and C ∞ (M ) the ring of smooth real-valued functions on M . Given v ∈ V (M ), we denote by φvt (x) an element of the local 1-parameter group of diffeomorphisms or flows generated by v through the point x ∈ M for sufficiently small t. Standard notations for Lie derivatives and Lie brackets are used which can be found in [6], [7]. Finally, we denote by 0k the zero vector with k elements and by Im the m × m identity matrix. Following [8], we denote the class of closed, connected embedded submanifolds of Rn which are controlled invariant for (1) by I (f, g, Rn ). If N ∈ I (f, g, Rn ), we write F (f, g, N ) for the collection of “friends” of N , i.e. maps u ¯ : N → Rm such that f + g u ¯ is tangent to N , i.e., (f + g u ¯)|N : N → T N. Given a distribution D on Rn , we let D⊥ be its orthogonal complement employing the natural orthogonal structure of Rn . This stands in contrast to the notation ann (D) which we use to denote the annihilator of D. If N ⊂ Rn is a submanifold and D a distribution on Rn , by T V +D is meant the vector bundle over V defined fibre-wise by Tp V + D(p); by ann(T V + D) ⊂ (T Rn )⋆ is meant the annihilator of ¯ T V + D over V . The involutive closure of D is denoted D. B. Problem Statement Suppose we are given a pair (Γ⋆ , u⋆ ), where Γ⋆ ∈ I (f, g, Rn ), dim Γ⋆ = n⋆ , and u⋆ ∈ F (f, g, Γ⋆ ). In this paper we investigate the following.

Local Transverse Feedback Linearization Problem (LTFLP): Given a point p0 ∈ Γ⋆ and a pair (Γ⋆ , u⋆ ), find conditions which ensure that there exist a neighborhood U of p0 in Rn , a local diffeomorphism Ξ : U → Ξ(U ), and a regular static feedback (α, β) such that, letting V := U ∩Γ⋆ , system (1) is feedback equivalent on U to a system modeled by equations of the form z˙ = f 0 (z, ξ) + g 1 (z, ξ)v1 + g 2 (z, ξ)v2 ξ˙ = Aξ + Bv1 ⋆

¯ i = G¯i . Lemma III.1 For all non-negative integers i, G ⋆ Thus, when Tp Γ = {0}, ρi = ri . The proof of Lemma III.1 is omitted for brevity. Inspired by [9], we conjecture that in cases when the dynamics of (1) transverse to Γ⋆ cannot be feedback linearized the transverse controllability indices characterize the largest transverse subsystem of (1) that can be feedback linearized.

(2)

⋆

IV. P RELIMINARY R ESULTS

where (z, ξ) ∈ Ξ(U ) ⊂ Rn ×Rn−n , v = col(v1 , v2 ) ∈ Rm , B is full rank, the pair (A, B) is controllable and Ξ|V is the canonical immersion Ξ|V :V → V × {0} z 7→ (z, 0).

Before proving the main results, we require some additional machinery which will generate a smooth feedback transformation which orders the vector fields of the distributions Gi in a convenient way. We begin with a necessary condition for LTFLP to be solvable.

Under mild regularity conditions, necessary and sufficient conditions for the existence of a solution to LTFLP are presented in Theorem IV.3.

Lemma IV.1 Suppose that LTFLP is solvable at p0 ∈ Γ⋆ . Then there exists a neighborhood U of p0 in Rn such that, letting V := Γ⋆ ∩ U , we have that (∀p ∈ V ) (∀i ∈ n − n⋆ )

III. T RANSVERSE C ONTROLLABILITY I NDICES

¯ i (p)) = dim(Tp V + Gi (p)) = const., dim(Tp V + G

In [1], we introduced the transverse controllability indices of (1) with respect to Γ⋆ . For our present discussion we define, for i = 0, 1, . . ., the following distributions associated with (1) Gi = span {adjf gk : 0 ≤ j ≤ i, 1 ≤ k ≤ m}, Gf = f + G0 = {f + g : g ∈ G0 } [∆, Λ] = {[X, Y ] : X ∈ ∆, Y ∈ Λ} (∆, Λ distributions), adif G0 = span{adif X : X ∈ G0 }, i = 0, 1, . . . , Gi = Gi−1 + [Gf , Gi−1 ], G0 = G0 , i = 1, 2, . . . , Si = G¯i−1 + adi G0 , S0 = G0 , i = 1, 2, . . . . f

In [1] the distributions Gi are assumed to be involutive in a neighborhood of Γ⋆ . Here we omit this assumption and generalize our definition of transverse controllability indices. For each p ∈ Γ⋆ , i ∈ N, define ρ0 (p) = dim (Tp Γ⋆ + G0 (p)) − n⋆ ¯ i−1 (p) + adif G0 (p)) ρi (p) = dim (Tp Γ⋆ + G ¯ i−1 (p)). − dim (Tp Γ⋆ + G When the integers ρi are constant over Γ⋆ (or an open subset of Γ⋆ ) we define a set of ρ0 integers k1 , . . . , kρ0 which are the transverse controllability indices of (1) with respect to Γ⋆ , ki := card {ρj ≥ i, j ≥ 0}, i ∈ ρ0 . When the Gi ’s are involutive, the definition of the integers ρi agrees with that given in [1]. In the special case when Γ⋆ is an equilibrium point, it is useful to compare our definition of to that given by Marino in [9]. There, controllability indices are defined using the integers r0 = dim G0 ri = dim Si − dim G¯i−1 in place of the integers ρi . The next result states that, when Tp Γ⋆ = {0}, the integers ρi and ri are identical.

which implies that ρ0 , . . . , ρn−n⋆ −1 are constant on V . Proof: Since the stated condition is coordinate and feedback independent it suffices to show that the lemma holds in (z, ξ) coordinates. By the properties of the normal form (2), for any p ∈ V , Ξ(p) = (p, 0) so that in (z, ξ) coordinates we have that for all p ∈ V Tp V + Gi (col(p, 0)) =  In⋆ ⋆ Im 0n−n⋆ ×n⋆ B

⋆ AB

··· ···

⋆ Ai B



(3)

for i ∈ n − n⋆ . It is clear from (3) that for all i ∈ n − n⋆ , the subspace Tp V + Gi (col(p, 0)) has constant dimension on ¯ i (col(p, 0)) ⊆ Tp V + V . We now show that, for all p ∈ V , G Gi (col(p, 0)). Once this is proven, the lemma follows from ¯ i (col(p, 0)). the fact that Tp V + Gi (col(p, 0)) ⊆ Tp V + G In (z, ξ) coordinates consider the constant distribution on U given by   In⋆ 0 0 ··· 0 ∆ = Im 0 B AB · · · Ai B where each column is a (constant) vector. At each p ∈ V , ∆(p) = Tp V + Gi (col(p, 0)). Since ∆ is involutive and ¯ i (col(p, 0)) ⊆ ∆(p) = Tp V + contains Gi , it follows that G Gi (col(p, 0)). The next result adapts [1, Lemma IV.1, Lemma IV.3] to the result in Lemma IV.1. In order to identify directions in the ¯ i (p) which are not contained in the intersection Tp V ∩ G ¯ intersection Tp V ∩ Gi−1 (p), define the integers ¯ 0 (p)) µ0 (p) := dim(Tp V ∩ G ¯ i (p)) − dim(Tp V ∩ G ¯ i−1 (p)) µi (p) := dim(Tp V ∩ G ni (p) :=

i X j=0

µj .

¯ i (p)). When the ρi ’s and µi ’s Thus, ni (p) = dim (Tp V ∩ G are constant over an open subset of Γ⋆ we have the following result. Lemma IV.2 Let W ⊆ Rn be an open set such that W ∩ Γ⋆ 6= ∅, and define V := W ∩ Γ⋆ . Assume that, for all i ∈ n − n⋆ , and for all p ∈ V ¯ i (p)) = constant dim (Tp V + Gi (p)) = dim (Tp V + G ¯ i (p)) = constant. (∀p ∈ W ) dim (G Then ρ0 ≤ ρ1 ≤ · · · ≤ ρn−n⋆ −1 k1 ≥ k2 ≥ · · · kρ0 . Moreover there exists an open set U ⊆ W , a regular static feedback (α, β) defined on U and ni vector fields vℓj : U → T Rn , 1 ≤ j ≤ i, 1 ≤ ℓ ≤ µi , such that, letting V˜ := U ∩ Γ⋆ 6= ∅, for all i ∈ n − n⋆ , k Gi (p)

(∀p ∈ V˜ )

:=

span{v10 , . . . , vµi i }(p)

⊆ Tp V˜

(∀p ∈ U ) ¯ i (p) = G

k Gi (p)

  i M span{adjf˜g˜k : 1 ≤ k ≤ ρj } ⊕ j=0

where f˜ = f + gα and g˜ = gβ.

¯ i by distinguishing between This lemma gives a basis of G ¯ the vector fields in Gi which are tangent to Γ∗ and those which are transverse to it. Specifically, we have that ¯ i = Gk ⊕ G⋔ (∀p ∈ U ) G i i     k k k ⋔ ⋔ + G + · · · G = G0 + G1/0 + · · · Gi/i−1 ⊕ G⋔ 0 1/0 i/i−1

k where Gi ⊆ T V and, for all p ∈ U , V

k

Gi/i−1 := span {vji : 1 ≤ j ≤ µi }

i G⋔ i/i−1 := span {adf gj : 1 ≤ j ≤ ρi } ⊆ Gi

(4)

span, respectively, the tangential and transversal directions ¯ i not contained in G ¯ i−1 . The proof of Lemma IV.2 is in G omitted since it is conceptually the same as the proof of [1, Lemma IV.1]. consequence of Lemma IV.2 is P An immediate that when ki = n − n⋆ , i.e., ¯ k −1 (p)) = n, (∀p ∈ V ) dim(Tp V + G 1

then, after feedback transformation, Tp V ⊕ span{adjf gk (p) : 0 ≤ j ≤ n − n⋆ − 1, 1 ≤ k ≤ ρj } = Tp Rn . (5) As a result, Lemma IV.2 yields the following array of n independent vector fields. In the array we use the symbols k Gi/i−1 and G⋔ i/i−1 to indicate a family of vector fields and

not the span of vector fields. ⋔ τ Gτ0 , G⋔ 0 ; . . . ; Gkρ0 −1/kρ0 −2 , Gkρ0 −1/kρ0 −2 ; Gτkρ /kρ −1 , G⋔ kρ0 /kρ0 −1 ; . . . . . . ; 0 0 Gτkρ −1 −1/kρ−1 −2 , G⋔ kρ0 −1 −1/kρ0 −1 −2 ; 0 · · · · · · · · · · τ ⋔ ρ0 − 1 Gτk3 /k3 −1 , G⋔ k3 /k3 −1 ; . . . ; Gk2 −1/k2 −2 , Gk2 −1/k2 −2 ; τ ⋔ τ ρ0 Gk2 /k2 −1 , Gk2 /k2 −1 ; . . . ; Gk1 −1/k1 −2 , G⋔ k1 −1/k1 −2 ; ρ0 + 1 v1 , , . . . , vn⋆ −nk1 −1 . (6) All of the vector fields of (6) are defined on U except for those in row ρ0 + 1. Those vector fields are solely defined on V ⊂ Γ∗ and are not contained in any of the ¯ i ’s so that at each p ∈ V , span{v1 , . . . , vn⋆ −n }(p) ≃ G k1 −1
¯ k −1 (p) /G ¯ k −1 (p). They are chosen to complete Tp V + G 1 1 the basis for Tp V , so that

1 2

k

Tp V = span{v1 , . . . , vn⋆ −nk1 −1 }(p) ⊕ Gk1 −1 (p). We conclude this section with the local version of [1, Theorem VI.1] which is used to prove our main result in Section V. Theorem IV.3 LTFLP is solvable at p0 if and only if there exist ρ0 smooth R-valued functions α1 , . . . , αρ0 , defined on some open neighborhood U of p0 in Rn , such that (1) U ∩ Γ⋆ ⊂ {x ∈ U : αi (x) = 0, i = 1, . . . , ρ0 } (2) The system x˙ = f (x) +

ρ0 X

gi (x)ui

i=1

′

(7)

y = col(α1 (x), . . . , αρ0 (x)) has vector relative degree {k1 , . . . , kρ0 } at p0 . V. M AIN R ESULT ¯ i , i ∈ k1 − 2 are regular at Theorem V.1 Assume that G p0 . Then LTFLP is solvable at p0 if and only if (a) dim (Tp0 Γ⋆ + Gk1 −1 (p0 )) = n and there exists an open neighborhood O of p0 in Γ⋆ such that (b) (∀i ∈ k1 − 2) (∀p ∈ O) dim (Tp Γ⋆ + Gi (p)) = ¯ i (p)) = constant. dim (Tp Γ⋆ + G Sketch of the Proof: (⇒) Assume LTFLP is solvable. Then condition (a) follows by [1, Lemma V.1]. Condition (b) follows by Lemma IV.1. (⇐) Conditions (a) and (b) along with the regularity of ¯ i imply that there exists a neighborhood W of p0 in Rn G such that the assumptions of Lemma IV.2 hold, and thus after feedback transformation we obtain the n independent vector fields of (6) defined on an open set U ⊆ W with V := U ∩ O 6= ∅. We now construct ρ0 R-valued functions α1 , . . . , αρ0 satisfying Theorem IV.3. Let p0 ∈ V be the origin for S-coordinates [10]. These coordinates are generated by composing the flows of the vector fields in (6) in a special order and then using the n flow times as coordinates. Scalar

functions α1 , . . . , αρ0 satisfying Theorem IV.3 are chosen from among those times. We compose the flows generated by the vector fields in (6) starting from the bottom row. Begin with the flows generated by the vector fields v1 , . . . , vn⋆ −nk1 −1 . Consider the mapping ⋆ F∅ : Ω ⊂ Rn −nk1 −1 → V ⊂ Rn . k

k

F∅ : S∅ := (s1 , . . . ,sn⋆ −nk 7→ φ

1 −1

)

vn⋆ −nk

◦ · · · ◦ φvk1 (p0 ).

1 −1

k

sn⋆ −n

s1

k1 −1

k

To each pair (Gi/i−1 , G⋔ i/i−1 ) in (6) we associate a set k

⋔ of times. For1 i ∈ k1 , let Si/i−1 = (Si/i−1 ; Si/i−1 ) := k

k

⋔ (s⋔ (i/i−1,1) , . . . , s(i/i−1,ρi ) ; s(i/i−1,1) , . . . , s(i/i−1,µi ) ). Next we generate a collection of mappings Fi/i−1 : Ui/i−1 ⊂ Rµi +ρi → Rn , (1 ≤ i ≤ k1 − 1), given by   k k ⋔ Fi/i−1 : Si/i−1 = Si/i−1 ; Si/i−1 7→ Φi/i−1 ◦ Φ⋔ i/i−1 (p), k

Φi/i−1 := φ

Φ⋔ i/i−1 := φ

i vµ

i k s(i/i−1,µ ) i

adif g1 s⋔ (i/i−1,ρi )

◦ ··· ◦ φ

◦ ··· ◦ φ

v1i

k

s(i/i−1,1)

adif gρi s⋔ (i/i−1,1)

,

k

⋔ The notation for Si/i−1 = (Si/i−1 ; Si/i−1 ) describes the fact ⋔ that Si/i−1 is a collection of times associated with vector ¯ i , not in G ¯ i−1 , which are transversal to V on fields in G k V . Meanwhile, Si/i−1 is a collection of times associated ¯ i , not in G ¯ i−1 , which are tangent to with vector fields in G V on V . Compose each of these maps together to obtain F : U ⊂ Rn → Rn ,

(8)

The fact that, for some sufficiently small neighborhood U of p0 in Rn , (8) is a diffeomorphism onto its image is an obvious consequence of property (a). Globally, i.e., in a neighborhood of Γ⋆ , it is not obvious. In the proof of Theorem III.1 in [1] we mistakenly claimed that (8) is a diffeomorphism from a neighborhood of Γ∗ onto its image. This mistake does not affect the conceptually similar, but significantly simpler, proof of Theorem 4.4 in [5], which provides sufficient conditions for the existence of a global solution to TFLP for single-input systems. The final S-coordinates are given by
S = col S∅ ; Sk1 −1/k1 −2 ; . . . ; S1/0 ; S0 .

As candidate output functions, let αi , i ∈ {1, . . . , ρ0 }, be the time spent flowing along adfki −1 gi , i.e. αi (x) = s⋔ (ki −1/ki −2,ρi −i+1) (x). With this choice for α, we must show that the conditions of Theorem IV.3 are satisfied. Re-define V as V = F (U ) ∩ Γ∗ . In [1, Theorem III.1] it is shown that V ⊆ {x : α(x) = 0} and that for all p ∈ F (U ) Ladℓf gj αi (p) = 0; 1 ≤ i ≤ ρ0 , 1 ≤ j ≤ m, 0 ≤ ℓ ≤ ki − 2. (9) 1 We

define i/(i-1) := 0 for i = 0 to be consistent with the array (6).

is full rank for any p ∈ V . Notice that for any point on V the last m − ρ0 columns of (10) are zero. To see this, recall that the preliminary feedback transformation of Lemma IV.2 is such that, (∀i ∈ {0, 1, . . .}) (∀p ∈ V ) (∀k ∈ {ρi + 1, . . . , m}) ¯ i−1 (p). adif gk (p) ∈ Tp V + G This implies, by the fact that V ⊆ {x : α(x) = 0} and by (9), that Ladki −1 g αi (p) = 0; 1 ≤ i ≤ ρ0 , i < j ≤ m. f

.

F := F0 ◦ F1/0 ◦ · · · ◦ Fk1 −1/k1 −2 ◦ F∅ (p0 ).

Since the proof of the above facts remains the same in the more general setting of this theorem, we focus on showing that the ρ0 × m decoupling matrix   Lgm Lfk1 −1 α1 (p) Lg1 Lfk1 −1 α1 (p) · · ·  L Lk2 −1 α (p) · · · Lgm Lfk2 −1 α2 (p)    2 g1 f   (10) ··· ··· ···   k −1 k −1 Lg1 Lf ρ0 αρ0 (p) · · · Lgm Lf ρ0 αρ0 (p)

j

Using [6, Lemma 4.1.2] allows us to set the final m − ρ0 columns of (10) to zero. Thus we concentrate on the ρ0 × ρ0 sub-matrix of (10) consisting of the first ρ0 columns. By [6, Lemma 4.1.2] and by (9) we have that the ρ0 ×ρ0 sub-matrix of (10) consisting of the first ρ0 columns is nonsingular if and only if   Ladk1 −1 gρ α1 (p) Ladk1 −1 g1 α1 (p) · · · f f 0    Ladk2 −1 g1 α2 (p) · · · Ladk2 −1 gρ α2 (p)   (11)  f f 0   ··· ··· ···   Ladkρ0 −1 g αρ0 (p) · · · Ladkρ0 −1 g αρ0 (p) f

1

f

ρ0

is non-singular. Now suppose k1 = k2 = · · · km1 > km1 +1 , 1 ≤ m1 ≤ ρ0 . Intuitively this means that the functions α1 , . . . , αm1 correspond to the times flowing along vector ¯ k −1 , but not in G ¯ k −2 . In terms of the ρi , this fields in G 1 1 means ρk1 −1 = ρk1 −2 = · · · = ρkm1 +1 = m1 . We now show that the first m1 rows of (11) are full rank. Suppose that there exist m1 scalars ci such that m1 X i=1

E D ci dαi , adfk1 −1 gj (p) = 0; 1 ≤ j ≤ ρ0 .

P ¯ k −1 ). However, This ci dαi ∈ ann (G 1 P implies that ci dαi ∈ ann (T V ). Therefore m1 X

¯ k −1 ) ci dαi ∈ ann (T V ) ∩ ann (G 1

i=1


¯ k −1 = 0. = ann T V + G 1

Since {dα1 , . . . , dαm1 } are linearly independent, (a fact easily seen in S-coordinates), we conclude that c1 = · · · = cm1 = 0 and the first m1 rows of (11) are full rank as claimed. Now suppose that km1 +1 = · · · = km1 +m2 > km1 +m2 +1 , 0 ≤ m2 ≤ ρ0 − m1 . We want to show that the first m1 + m2

rows of (11) are full rank. In order to do this we first show that the exact one-forms dLif αj , 1 ≤ j ≤ m1 , 0 ≤ i ≤ k1 − km1 +1 − 1 are (i) linearly independent on V .
¯k (ii) Contained in ann T V + G . m1 +1 −2 Fact (ii) follows directly from (9) and the fact that V ⊆ {x : α(x) = 0}. To prove (i), consider the linear combination m1 X

where h = m1 (k1 − km1 +1 ) + nkm1 +1 −1 and k = n − m1 (k1 −km1 +1 +1)−m2 −nkm1 +1 −1 . In light of this the term Pm1 k1 −km1 +1 αi in (14) must have, in S-coordinates, i=1 ai dLf the form   −b1 · · · −bm2 0m1 0k . ⋆h

¯k However, in S-coordinates, vector fields in G have m1 +1 −1 the form col(0h , ⋆) with zeros corresponding precisely with the term ⋆h above. In fact it is possible to find a v ∈ ¯k ¯ G −1 , v 6∈ Gk −2 , given by m1 +1

a0i dαi

+ ··· +

k −k k −k ai 1 m1 +1 dLf 1 m1 +1 αi

= 0.

(12)

i=1

⇔

k −km1 +1

ai 1

dαi , adfk1 −1 gj

i=1

+

= 0.

¯k ann(T V +G )= m1 +1 −1 M

span {dLif α1 , . . . , dLif αm1 }.

(13)

i=0

Returning our attention to the first m1 + m2 rows of (11), suppose there exist m1 +m2 scalars such that for 1 ≤ j ≤ ρ0 m1 X i=1

D E ai dαi , adfk1 −1 gj +

m2 X i=1

Using, once again [6, Lemma 4.1.2] and (9) this can be written as + *m m2 1 X X km1 +1 −1 k1 −km1 +1 gj = 0. bi dαm1 +i , adf αi + ai dLf i=1

which implies that at each p ∈ V m1 X i=1

k −km1 +1

ai (p)dLf 1

αi (p) +

k

adf m1 +1

−1

gi

0h

v = col

This means that *m 1 X

···

0

0

⋆m1

k −k ai dLf 1 m1 +1 αi , v

+

0k




.

=0

and hence + *m km1 +1 −1 1 X X k1 −km1 +1 km1 +1 −1 km1 +1 −1 αi , ai dLf gi = 0. adf ci i=1

i=0

k1 −km1 +1 αi i=1 ai dLf

Pm 1

¯k ∈ ann (T V + G ) Thus m1 +1 −1 which by the fact (i) shown earlier, implies that ai ≡ 0. We are left to show that the bi in (14) are zero. This can be done directly using (9) and [6, Lemma 4.1.2] and considering the expression m1 X k −k −1 k −k −1 a0i dαi + · · · + ai 1 m1 +1 dLf 1 m1 +1 αi + i=1

m2 X

bi dαm1 +i = 0.

i=1

D E k −1 bi dαm1 +i , adf m1 +1 gj = 0.

i=1

−1

such that in S-coordinates

i=1

Since the first m1 rows of (11) are linearly independent, we k −k conclude that ai 1 m1 +1 = 0 for 1 ≤ i ≤ m1 . Following this same procedure one can recursively show that all the coefficients in (12) are identically zero and (i) is proven. An important consequence of this fact is that k1 −km1 +1 −1

k

c0i gi + · · · + ci m1 +1

i=0

−1

Next take the inner product of (12) with adf m1 +1 gj , 1 ≤ j ≤ m. Using (9) and [6, Lemma 4.1.2] we have + *m 1 X k1 −km1 +1 −1 k1 −km1 +1 k1 −km1 +1 gj = 0 αi , adf dLf ai *m 1 X

X

v=

i=1

k

m1 +1

km1 +1 −1

m2 X

bi (p)dαm1 +i (p)

(14)

i=1

¯k belongs to ann (T V + G ). We now show that this m1 +1 −1 contradicts (13). Consider the S-coordinates representation Pm 2 of the term b dα in (14). Since the one-forms m1 +i i=1 i {dαm1 +1 , . . . , dαm1 +m2 } are part of the dual basis in Scoordinates, it has a particularly simple vector notation given by   b1 · · · b m 2 0m1 0h 0k

One now proceeds in exactly the same way as was used to show that the coefficients in (12) are all zero. At this point the proof technique can be repeated until all the rows of (11) are accounted for. Specifically, the next step in the proof is to assume that km1 +m2 +1 = · · · = km1 +m2 +m3 > km1 +m2 +m3 +1 . Now take a linear combination of the first m1 + m2 + m3 rows of (11) and assume there exists m1 + m2 + m3 scalars such that, for 1 ≤ j ≤ ρ0 , m2 m1 E D E X D X k −1 bi dαm1 +i , adf m1 +1 gj ai dαi , adfk1 −1 gj + i=1 m3 X

+

i=1

i=1

D

k

ci dαm1 +m2 +i , adf m1 +m2 +1

−1

E gj = 0.

Arguing in the same way as above, one shows that the integers ai , bi , and ci must be identically zero. In this way one shows that (11) is full rank. In conclusion, the function (α1 , . . . , αρ0 ) constructed using S-coordinates satisfy both conditions in Theorem IV.3.

VI. E XAMPLE To illustrate these ideas we present an example of centralized control of two Lorenz oscillators. The equations of motion are x˙ 1 = σ(x2 − x1 ) + u1 y˙ 1 = σ(y2 − y1 ) + u2 x˙ 2 = rx1 − x2 − x1 x3 x˙ 3 = −bx3 + x1 x2 + u1

y˙ 2 = ry1 − y2 − y1 y3 (15) y˙ 3 = −by3 + y1 y2 + u2 .

For simplicity, we assume that σ = r = b = 1. We consider two separate problems: (a) the problem of full state synchronization and (b) a partial synchronization problem. We will show that the latter is solvable while the former is not by using transverse feedback linearization. These types of problems are common and have appeared in the literature [11]. We begin with the partial synchronization problem. Suppose we are interested in forcing the the variables x1 and y1 to lie on a unit circle Γ⋆ = {(x, y) ∈ R3 × R3 : x21 + y12 = 1}. In this case it is clear that u⋆ = col(−σ(x2 − x1 ) + y1 , −σ(y2 − y1 ) − x1 ) is a suitable, though not unique, friend. The constraint h = x21 + y12 − 1 defining Γ⋆ satisfies condition (1) of Theorem IV.3. It turns out that condition (2) holds as well signifying that the constraint can be used as the virtual output y ′ in (7). It is instructive to also check the conditions of Theorem V.1. In this example we have that for all x ∈ Γ⋆ ,   0 0 0 0 y1  1 0 0 0 0     0 1 0 0 0  ⋆  . Tx Γ = Im    0 0 0 0 −x1   0 0 1 0 0  0 0 0 1 0 Simple calculations reveal that for all x ∈ Γ⋆ , dim(Tx Γ⋆ + G0 (x)) = 6, i.e. condition (a) of Theorem V.1 is satisfied. In the special case when n⋆ = n − 1, the conditions of Theorem V.1 simplify and condition (a) becomes both necessary and sufficient.Following the procedure in the proof of Theorem IV.3 one obtains the output function  q x21 + y12 . α(x, y) = ln

Next we pursue the question of whether or not transverse feedback linearization can be used to synchronize system (15), i.e we want to know if the diagonal Γ⋆ = {(x, y) ∈ R3 × R3 : x1 = y1 , x2 = y2 , x3 = y3 }

can be stabilized using transverse feedback linearization. The set Γ⋆ is invariant for any choice of u⋆ so long as u⋆1 = u⋆2 . In particular u⋆ = 0 is a suitable candidate. Thus n⋆ = 3 and for any p ∈ Γ⋆ we have that   1 0 0  0 1 0     0 0 1  ⋆ .  Tp Γ = Im    1 0 0   0 1 0  0 0 1

To check that conditions of theorem V.1 we note that G0 = ¯ 0 and that for all p ∈ Γ⋆ dim(Tp Γ⋆ + G0 (p)) = 4 which G means that ρ0 = 1. Next we find the distribution G1 by noting that adf g1 (x, y) = col(1, x1 + x3 − 1, 1 − x2 , 0, 0, 0) and adf g2 (x, y) = col(0, 0, 0, 1, y1 + y3 − 1, 1 − y2 ). Simple calculations give that ρ1 = 1 and so for condition (b) of Theorem V.1 to hold we require that for all p ∈ Γ⋆ , ¯ 1 (p)). One can easily dim(Tp Γ⋆ + G1 (p)) = dim(Tp Γ⋆ + G check that this condition fails and hence the conditions of Theorem V.1 do not hold. We conclude that transverse feedback linearization cannot be used to synchronize the Lorenz oscillators (15). VII. CONCLUSIONS Together with earlier results in [5] and [1], this paper completes the characterization of the local transverse feedback linearization problem. The main contributions of this paper are: a generalization of the definition of controllability indices which can be used when the distributions Gi are not involutive; the comparison between our definition of transverse controllability indices in the special case when Tp Γ⋆ = {0} and Marino’s definition (Lemma III.1); a new necessary condition (Lemma IV.1) and checkable necessary and sufficient conditions (Theorem V.1) for the existence of a solution to LTFLP. Future research includes solving the global version of TFLP. R EFERENCES [1] C. Nielsen and M. Maggiore, “Transverse feedback linearization of multi-input systems,” in IEEE Conference on Decision and Control and European Control Conference, Seville, December 2005, pp. 4897– 4902. [2] A. Banaszuk and J. Hauser, “Feedback linearization of transverse dynamics for periodic orbits,” Systems and Control Letters, vol. 26, no. 2, pp. 95–105, Sept. 1995. [3] B. Morris and J. Grizzle, “A restricted Poincar´e map for determining exponentially stable periodic orbits in systems with impulse effects: Application to bipedal robots,” in IEEE Conference on Decision and Control and European Control Conference, Seville, December 2005, pp. 4199–4206. [4] A. Shiriaev, J. Perram, and C. de Wit, “Constructive tool for orbital stabilization of underactuated nonlinear systems: Virtual constraints approach,” IEEE Transactions on Automatic Control., vol. 50, no. 8, pp. 1164–1176, August 2005. [5] C. Nielsen and M. Maggiore, “Output stabilization and maneuver regulation: A geometric approach,” Systems Control Letters, vol. 55, no. 5, pp. 418–427, 2006. [6] A. Isidori, Nonlinear Control Systems, 3rd ed. New York: SpringerVerlag, 1995. [7] H. Nijmeijer and A. van der Schaft, Nonliner Dynamical Control Systems. New York: Springer-Verlag, 1990. [8] W. Wonham, Linear Multivariable Control: A Geometric Approach., 3rd ed. New York: Springer-Verlag, 1985. [9] R. Marino, “On the largest feedback linearizable subsystem,” Systems and Control Letters, vol. 6, no. 5, pp. 345–351, 1986. [10] R. Su and L. Hunt, “A canonical expansion for nonlinear systems,” IEEE Transactions on Automatic Control., vol. 31, no. 7, pp. 670–673, 1986. [11] A. Pogromsky, G. Santoboni, and H. Nijmeijer, “Partial synchronization: from symmetry towards stability,” Phys. D, vol. 172, no. 1-4, pp. 65–87, 2002.